{"id":1444,"date":"2018-02-06T16:50:06","date_gmt":"2018-02-06T16:50:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&p=1444"},"modified":"2018-02-19T15:52:22","modified_gmt":"2018-02-19T15:52:22","slug":"9-chapter-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/9-chapter-review\/","title":{"raw":"9 Chapter Review","rendered":"9 Chapter Review"},"content":{"raw":"
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Key Terms<\/span><\/h3>\r\n
\r\n \t
center of mass<\/strong><\/dt>\r\n \t
weighted average position of the mass<\/dd>\r\n<\/dl>\r\n
\r\n \t
closed system<\/strong><\/dt>\r\n \t
system for which the mass is constant and the net external force on the system is zero<\/dd>\r\n<\/dl>\r\n
\r\n \t
elastic<\/strong><\/dt>\r\n \t
collision that conserves kinetic energy<\/dd>\r\n<\/dl>\r\n
\r\n \t
explosion<\/strong><\/dt>\r\n \t
single object breaks up into multiple objects; kinetic energy is not conserved in explosions<\/dd>\r\n<\/dl>\r\n
\r\n \t
external force<\/strong><\/dt>\r\n \t
force applied to an extended object that changes the momentum of the extended object as a whole<\/dd>\r\n<\/dl>\r\n
\r\n \t
impulse<\/strong><\/dt>\r\n \t
effect of applying a force on a system for a time interval; this time interval is usually small, but does not have to be<\/dd>\r\n<\/dl>\r\n
\r\n \t
impulse-momentum theorem<\/strong><\/dt>\r\n \t
change of momentum of a system is equal to the impulse applied to the system<\/dd>\r\n<\/dl>\r\n
\r\n \t
inelastic<\/strong><\/dt>\r\n \t
collision that does not conserve kinetic energy<\/dd>\r\n<\/dl>\r\n
\r\n \t
internal force<\/strong><\/dt>\r\n \t
force that the simple particles that make up an extended object exert on each other. Internal forces can be attractive or repulsive<\/dd>\r\n<\/dl>\r\n
\r\n \t
Law of Conservation of Momentum<\/strong><\/dt>\r\n \t
total momentum of a closed system cannot change<\/dd>\r\n<\/dl>\r\n
\r\n \t
linear mass density<\/strong><\/dt>\r\n \t
\u03bb<\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>, expressed as the number of kilograms of material per meter<\/dd>\r\n<\/dl>\r\n
\r\n \t
momentum<\/strong><\/dt>\r\n \t
measure of the quantity of motion that an object has; it takes into account both how fast the object is moving, and its mass; specifically, it is the product of mass and velocity; it is a vector quantity<\/dd>\r\n<\/dl>\r\n
\r\n \t
perfectly inelastic<\/strong><\/dt>\r\n \t
collision after which all objects are motionless, the final kinetic energy is zero, and the loss of kinetic energy is a maximum<\/dd>\r\n<\/dl>\r\n
\r\n \t
rocket equation<\/strong><\/dt>\r\n \t
derived by the Soviet physicist Konstantin Tsiolkovsky in 1897, it gives us the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass from\u00a0m<\/span>i<\/span><\/span><\/span><\/span><\/span><\/span><\/span>mi<\/span><\/span>\u00a0down to\u00a0m<\/em><\/dd>\r\n<\/dl>\r\n
\r\n \t
system<\/strong><\/dt>\r\n \t
object or collection of objects whose motion is currently under investigation; however, your system is defined at the start of the problem, you must keep that definition for the entire problem<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>\r\n
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Key Equations<\/span><\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Definition of momentum<\/td>\r\np<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>m<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>p\u2192=mv\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Impulse<\/td>\r\nJ<\/span>\u20d7\u00a0<\/span><\/span>\u2261<\/span>\u222b<\/span>t<\/span>f<\/span><\/span><\/span>t<\/span>i<\/span><\/span><\/span><\/span>F<\/span>\u20d7\u00a0<\/span><\/span>(<\/span>t<\/span>)<\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span>or<\/span><\/span>J<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>F<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>ave<\/span><\/span><\/span>\u0394<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span>J\u2192\u2261\u222btitfF\u2192(t)dtorJ\u2192=F\u2192ave\u0394t<\/span><\/span><\/td>\r\n<\/tr>\r\n
Impulse-momentum theorem<\/td>\r\nJ<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>\u0394<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>J\u2192=\u0394p\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Average force from momentum<\/td>\r\nF<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>\u0394<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u0394<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>F\u2192=\u0394p\u2192\u0394t<\/span><\/span><\/td>\r\n<\/tr>\r\n
Instantaneous force from momentum\r\n
<\/div>\r\n(Newton\u2019s second law)<\/td>\r\n
F<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>F\u2192(t)=dp\u2192dt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Conservation of momentum<\/td>\r\nd<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span>+<\/span>d<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span>=<\/span>0<\/span><\/span><\/span>or<\/span><\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>+<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>=<\/span>constant<\/span><\/span><\/span><\/span><\/span><\/span>dp\u21921dt+dp\u21922dt=0orp\u21921+p\u21922=constant<\/span><\/span><\/td>\r\n<\/tr>\r\n
Generalized conservation of momentum<\/td>\r\n\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span>constant<\/span><\/span><\/span><\/span><\/span><\/span>\u2211j=1Np\u2192j=constant<\/span><\/span><\/td>\r\n<\/tr>\r\n
Conservation of momentum in two dimensions<\/td>\r\np<\/span>f<\/span>,<\/span>x<\/span><\/span><\/span>=<\/span>p<\/span>1,i<\/span>,<\/span>x<\/span><\/span><\/span>+<\/span>p<\/span>2,i<\/span>,<\/span>x<\/span><\/span><\/span><\/span><\/span>p<\/span>f<\/span>,<\/span>y<\/span><\/span><\/span>=<\/span>p<\/span>1,i<\/span>,<\/span>y<\/span><\/span><\/span>+<\/span>p<\/span>2,i<\/span>,<\/span>y<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>pf,x=p1,i,x+p2,i,xpf,y=p1,i,y+p2,i,y<\/span><\/span><\/td>\r\n<\/tr>\r\n
External forces<\/td>\r\nF<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>ext<\/span><\/span><\/span>=<\/span>\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>d<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>F\u2192ext=\u2211j=1Ndp\u2192jdt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Newton\u2019s second law for an extended object<\/td>\r\nF<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>d<\/span>p<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>CM<\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>F\u2192=dp\u2192CMdt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Acceleration of the center of mass<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>CM<\/span><\/span><\/span>=<\/span>d<\/span>2<\/span><\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span>(<\/span>1<\/span>M<\/span><\/span>\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>m<\/span>j<\/span><\/span><\/span><\/span><\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span>)<\/span><\/span>=<\/span>1<\/span>M<\/span><\/span>\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>m<\/span>j<\/span><\/span><\/span><\/span><\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192CM=d2dt2(1M\u2211j=1Nmjr\u2192j)=1M\u2211j=1Nmja\u2192j<\/span><\/span><\/td>\r\n<\/tr>\r\n
Position of the center of mass for a system\r\n
<\/div>\r\nof particles<\/td>\r\n
r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>CM<\/span><\/span><\/span>\u2261<\/span>1<\/span>M<\/span><\/span>\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>m<\/span>j<\/span><\/span><\/span><\/span><\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span><\/span><\/span><\/span><\/span>r\u2192CM\u22611M\u2211j=1Nmjr\u2192j<\/span><\/span><\/td>\r\n<\/tr>\r\n
Velocity of the center of mass<\/td>\r\nv<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>CM<\/span><\/span><\/span>=<\/span>d<\/span>d<\/span>t<\/span><\/span><\/span>(<\/span>1<\/span>M<\/span><\/span>\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>m<\/span>j<\/span><\/span><\/span><\/span><\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span>)<\/span><\/span>=<\/span>1<\/span>M<\/span><\/span>\u2211<\/span>j<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>m<\/span>j<\/span><\/span><\/span><\/span><\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>j<\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192CM=ddt(1M\u2211j=1Nmjr\u2192j)=1M\u2211j=1Nmjv\u2192j<\/span><\/span><\/td>\r\n<\/tr>\r\n
Position of the center of mass of a\r\n
<\/div>\r\ncontinuous object<\/td>\r\n
r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>CM<\/span><\/span><\/span>\u2261<\/span>1<\/span>M<\/span><\/span>\u222b<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>m<\/span><\/span><\/span><\/span><\/span><\/span>r\u2192CM\u22611M\u222br\u2192dm<\/span><\/span><\/td>\r\n<\/tr>\r\n
Rocket equation<\/td>\r\n\u0394<\/span>v<\/span>=<\/span>u<\/span><\/span>ln<\/span>(<\/span>m<\/span>i<\/span><\/span><\/span>m<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u0394v=uln(mim)<\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n<\/div>\r\n
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Summary<\/span><\/h3>\r\n
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9.1<\/span>\u00a0<\/span>Linear Momentum<\/span><\/h4>\r\n